nLab strict terminal object



Category theory

Limits and colimits



The trivial ring, among all unital rings, has two characteristic properties:

  1. there is a unique function into the trivial ring, from any other unital ring;

  2. there is no function from the trivial ring, except to itself.

The first property generalizes to arbitrary categories as the property of a terminal object.

The corresponding generalization including also the second property is that of a strict terminal object:


A terminal object 1\mathbf{1} is called a strict terminal object if every morphism from 1\mathbf{1} is an isomorphism:

1X. \mathbf{1} \overset{\simeq}{\longrightarrow} \X \,.

In other words, a strict terminal object is a maximal terminal object.


  • Trivial unital ring

  • Trivial Boolean algebra

  • Trivial absorption monoid

  • In general, for any algebraic theory with two constants 00 and 11 and a binary operation for which 00 is a (left) absorbing element and 11 is a (left) unit, the trivial model is strictly terminal.


Last revised on March 17, 2023 at 15:01:08. See the history of this page for a list of all contributions to it.